doi:10.3934/ipi.2017038
Inverse Problems and Imaging
DATA DRIVEN RECOVERY OF LOCAL VOLATILITY SURFACES
Vinicius Albani Dept. of Mathematics, UFSC, Florianopolis, Brazil
Uri M. Ascher Dept. of Computer Science, University of British Columbia, Canada
Xu Yang and Jorge P. Zubelli IMPA, Rio de Janeiro, Brazil
(Communicated by Otmar Scherzer) Abstract. This paper examines issues of data completion and location uncertainty, popular in many practical PDE-based inverse problems, in the context of option calibration via recovery of local volatility surfaces. While real data is usually more accessible for this application than for many others, the data is often given only at a restricted set of locations. We show that attempts to “complete missing data” by approximation or interpolation, proposed and applied in the literature, may produce results that are inferior to treating the data as scarce. Furthermore, model uncertainties may arise which translate to uncertainty in data locations, and we show how a model-based adjustment of the asset price may prove advantageous in such situations. We further compare a carefully calibrated Tikhonov-type regularization approach against a similarly adapted EnKF method, in an attempt to fine-tune the data assimilation process. The EnKF method offers reassurance as a different method for assessing the solution in a problem where information about the true solution is difficult to come by. However, additional advantage in the latter approach turns out to be limited in our context.
1. Data manipulation and local volatility surfaces. The basic setup of data assimilation and inverse problems for model calibration consists of an assimilation of dynamics, defined for instance as a discretized PDE, and observed data [27, 26]. The celebrated Kalman filter, for example, does a forward pass on a weighted least squares problem fitting both model dynamics and data, and it guarantees variance minimization for a linear problem with Gaussian noise. However, in practice, the “statistical sanctity” of the data is often violated before the assimilation process commences. This can happen for various reasons and in different circumstances: 2010 Mathematics Subject Classification. Primary: 45Q05, 97M30, 65R32. Key words and phrases. Data science, local volatility calibration, inverse problem, Tikhonovtype regularization, ensemble Kalman filter. VA acknowledges and thanks CNPq through grant 201644/2014-2. UMA and XY acknowledge with thanks a Ciencias Sem Fronteiras (visiting scientist / postdoc) grant from CAPES, Brazil. JPZ thanks the support of CNPq grant 307873 and FAPERJ grant 201.288/2014. 1
c
200X American Institute of Mathematical Sciences
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Vinicius Albani, Uri M. Ascher, Xu Yang and Jorge P. Zubelli
1. In cases where the data is scarce, in the sense that it is observed only at a small set of locations compared to the size of a reasonable discretization mesh of a physical domain, there would be many models (solutions to the inverse problem) that explain the data (e.g., [16]). It is then tempting to “complete” the data by some interpolation or other approximation, whereupon the role of an ensuing regularization as a prior is less crucial. 2. There may be a hidden uncertainty in the locations of data, not only in data values (e.g., [19, 15]). For instance, engineers often prefer to see data given at regular mesh nodes, so a quiet constant interpolation, moving data items to the nearest cell vertex, is common practice. 3. Data completion may be necessary to obtain a more efficient algorithm [28, 25]. 4. A quiet data completion is often assumed by mathematicians in order to enable building theory for inverse problems. This includes assumptions of available data on continuous boundary segments [13], or of observed (measured) relationships between unknown functions that are presumed to hold everywhere in a physical domain. 5. There are situations where some form of data completion and other manipulation is necessary because no one knows how to solve the problem otherwise [25]. These observations raise the following questions: (i) when (and in what sense) is it practically acceptable to perform such data manipulations? (ii) in which circumstances can one gain advantage by treating the observed data more carefully? and (iii) how should one assess correctness of a solution that has been obtained with such manipulated data? Our general observation is that researchers occasionally, but not always, seem to get away with such “crimes”, in the sense of producing agreeable results. For instance, in [28] the authors obtained agreeable reconstructions so long as the percentage of completed data did not exceed about 50%, but not more. Such empirical evidence is relatively rare in the literature, however, and it depends on the problem at hand. More insight is therefore required, and such may be gained by considering applied case studies. In this article we focus on a model calibration problem in a setting that features both scarce data and uncertainty regarding data location. Further, it allows us to work with real rather than synthetic data, often available through the internet. This problem, which has had tremendous impact in mathematical finance, concerns the determination of the so-called local volatility surface, making use of derivative prices. A good model for the volatility is crucial for many applications ranging from risk management to hedging and pricing of financial instruments. The classical Black-Scholes-Merton model had subsumed a constant volatility model σ [6] in a simplifying stochastic dynamics for the underlying process [24]. However, the constant volatility assumption was quickly contradicted by the actual derivative prices observed in the market. The disagreement between the BlackScholes model-implied prices for different expiration dates and negotiated strike prices became known as the smile effect. A number of practical parametric as well as nonparametric models have been proposed in this context; see [14] and references therein. The parametric ones try to fulfill different phenomenological features of the observed prices. Yet, in a ground breaking paper Dupire [11] proposed the use of a function σ that depends on time and the price at that time. For the case of the European call contracts, he replaced the Black-Scholes equation by a PDE of the form Inverse Problems and Imaging
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(1)
∂C ∂τ
=
∂C 1 2 ∂2C − bK σ (τ, K)K 2 , 2 ∂K 2 ∂K
3
τ > 0, K ≥ 0,
with initial and boundary conditions (for calls) given by (2)
C(τ = 0, K)
=
(S0 − K)+ ,
lim C(τ, K)
=
0,
lim C(τ, K)
= S0 ,
K→∞
K→0
where τ is time to maturity, K is the strike price, and C = C(τ, K) is the value of the European call option with expiration date T = τ . The parameter S0 is the asset’s price at a given date. One ensuing complication in using (1), however, is in the calibration of this model, by which we mean finding a plausible volatility surface σ(τ, K) that matches, or explains, given market data on call option values. The task here is significantly more challenging than in the case where σ is a constant. This paper deals with the computational challenges that this inverse, or inference problem gives rise to, focussing in particular on the treatment of scarce data and uncertainty in data locations. We show that, contrary to a popular approach in the literature, avoiding data completion is the way to go here. Further, taking data location uncertainty into account, rather than simply ignoring it, improves reconstruction quality. The forward problem involves finding the values of C satisfying the differential problem (1)-(2) for given σ(τ, K) and S0 , evaluated at the points (τ, K) where data values are available. A major difficulty here is that the data are scarce. To explain what we mean by this, suppose we have discretized the PDE using, say, the Crank-Nicolson method on a rectangular mesh that is reasonable in the sense that the essence of the differential solution is retained in the discrete solution. Then the data is scarce in that the number M of degrees of freedom in the discrete C typically far exceeds the number of given data values l: l M . Moreover, the available data in some typical situations are given at locations that are far from the boundaries of the (truncated) domain on which the approximated PDE problem is solved. See Figures 1 and 5 below for examples of such (real) data sets. Now, if the local volatility surface is discretized, or injected, on the same mesh as that of the forward problem, then there are roughly M degrees of freedom in σ, which is again potentially far larger than the number of data constraints. We can of course discretize σ on a coarser sub-mesh (which in the extreme case would have only one point, thus leading back to a constant volatility), or parameterize the surface in a more involved manner; see [18, 17, 1, 12, 7, 2, 10, 4] and references therein for further detail. Here, however, we stick to a straightforward nodal representation of this surface on the full C-mesh in the hope of retaining flexibility and detail, while avoiding artifacts that may arise from restrictive simplifying assumptions. This approach has worked well in geophysical exploration problems [16], among others. Thus, the problem of finding a volatility surface that explains the data is often significantly under-constrained in practice. This does not make it easy to solve, however, as the ultimate goal is to obtain plausible volatility surfaces that can be worked with, and not just to match data. Our task is therefore to assimilate the data information with the information contained in the PDE model (1), using any plausible a priori information as a prior in the assimilation process. Such a priori information can vary significantly, addressing concerns of adherence to the financial Inverse Problems and Imaging
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model, relative smoothness of the volatility surface, and numerical stability issues, among others. One approach that has been relatively popular in financial circles is to apply to the data special interpolation/extrapolation methods that take into account the a priori information of the financial model (e.g., [23]). This is used to obtain data values at all points of the rectangular mesh on which C is defined, and subsequently the “new data” is assimilated with the information that the solutions of the Dupire equation for different σ’s yield to calibrate (1). An advantage with this data completion approach is that the data is no longer scarce when we get to the redefined inverse problem. This allows for developing some existence and uniqueness theory as well; see [9, 1], and references therein. However, a disadvantage is that such data completion constitutes a “statistical crime”, as the errors in the new data may no longer be considered as independent random variables, see [28]. In fact, we get two solutions C that are in a sense competing rather than completing one another, since the one satisfying (1), even for the “best” σ, does not necessarily satisfy the data interpolation conditions and vice versa. In Sections 2 and 5 we therefore examine the performance of this data completion approach against that of a scarce data approach that is based on a carefully tuned Tikhonov-type regularization. We have verified the robustness of our regularization operator by applying also variants of EnKF-like algorithms [22, 20, 8], further described below, obtaining similar results. Using both synthetic and real data sets, we show that the scarce data approach can give better and more reliable results; in our reported experiments this has happened especially for the real data applications. We then continue with the scarce data approach. The maximum a posteriori (MAP) functional considered in Section 2 is based on the statistical assumption that the data error covariance matrix is a scalar multiple of the identity. In Section 3 we subsequently consider an algorithm, based on an approach recently proposed in [20] and [22], where we attempt to learn more about the error covariance matrix as the iterative process progresses, using ensemble Kalman filter (EnKF) techniques. Although our problem is time-dependent, the time variable here does not really differ from the other independent variable in the usual sense. In particular, the unknown surface σ depends on both K and τ , unlike for instance the material functions in [20, 8, 16], which are independent of time. Thus, the EnKF-like methods considered use an artificial time [5]. In Section 3 we find that the EnKF algorithm can be improved in our context by adding smoothing prior penalties, just like in Section 2. The probabilistic setup, although general, is not fully effective as a substitute for prior knowledge that is available in no uncertain terms. The problem setting used in Sections 2 and 3 regards the asset price S0 as a known parameter. However, in practice there is uncertainty in this parameter. In fact, we have an observed value which is in the best case an average over a day of trading, so S0 should be treated as an unknown with an observed mean value and a variance that is relatively easy to estimate. This in turn affects the calibration problem and its solution process. Section 4 deals with this additional complication, which translates into uncertainty in the data locations of a transformed formulation for (1). In Section 5 we collect our numerical tests, addressing and assessing the various aspects of the methods desribed earlier. We use synthetic data in Section 5.1 to show the advantage in applying the method of Section 4 for problems with uncertainty in Inverse Problems and Imaging
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the price S0 . In Section 5.2 we use market equity data to fine-tune our regularization functional, as well as to compare Tikhonov-type regularization vs the modified artificial time EnKF. In Section 5.3 we use oil and gas commodity market data to further investigate data completion approaches, showing that the scarce data approach is superior. Conclusions are offered in Section 6. 2. Two approaches for handling scarce data. Below we assume that the parameter S0 is given.1 This assumption will be modified in Section 4. We then apply a standard transformation changing the independent variable K to the so-called log moneyness variable y = log(K/S0 ). This is followed by changing the dependent variables of the forward and inverse problems to u(τ, y) = C(τ, S0 exp(y)) and a(τ, y) = 21 σ(τ, K(y))2 , respectively. We obtain the dimension-less parabolic PDE with no unbounded coefficients 2 ∂ u ∂u ∂u ∂u (3) +a − +b = 0, τ > 0, y ∈ 0.5 (the so-called in-the-money and out-of-the-money regions), if we do not add the a0 penalty, the two wings blow up.
Figure 7. Reconstructed SPX local volatility surfaces at different maturities obtained with Tikhonov-type and EnKF methods using completed data. These results are inferior to the corresponding ones for scarce data, displayed in Figure 6. Figure 8 presents reconstructed SPX local volatility surfaces obtained with all six method variants. When put together, we can see that the different methods lead to results that get closer as the maturity gets longer around y = 0. Inverse Problems and Imaging
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Figure 8. Reconstructed SPX local volatility surfaces obtained with six method variants. See legends in Figures 6, 7 and 10. Figure 9 is a zoom-in of Figure 8 to the region around y = 0. In this region one expects a larger number of liquid contracts. Observe that the plotted curves are generally divided into two groups: one is obtained from the original (scarce) data and the other from the completed data. This phenomenon is clearer in the figures for the earliest and latest dates T = 25 and T = 332.
Figure 9. Reconstructed SPX local volatility surfaces obtained with six method variants for different maturities in the at-themoney (y = 0) neighbourhood. Taken together, Figures 8 and 9 again demonstrate the superiority of the scarce data approach, as well as the regularization methods that employ α1 > 0. One concept that is prevalent in practical applications in the so-called implied volatility. It is defined as the volatility that would be observed for a standard call (or put) contract to give the observed price if the classical Black-Scholes formula were used. In other words, it is the constant volatility that when plugged into (1) - all else being equal - would yield the corresponding quoted price. In Figure 10 we see that the reconstructions of implied volatility from local volatility do better for intermediate term and long term maturities. However, when the maturities are short, the results split into two groups according to whether we use the original scarce data or the completed data. It is important to emphasize that the red curves in Figure 10 represent implied volatility, not the prices (i.e., the red curves are not simple “observations” to be fitted). In financial markets, a misfit such as is displayed here for the short maturities may or may not be acceptable depending on issues such as bid and ask spread as well as risk premia and absence of arbitrage opportunities. Moreover, the simple fact that a problem is under-determined does not mean that we should be able to fit it well with a given model such as (1). In fact, even if such fitting were possible in a different (e.g., interpolation) sense, the prices obtained could be subject to arbitrage opportunities, and here indeed the practical use would be excluded. (There is incidentally a well-known and somewhat similar effect when over-fitting learning examples in machine learning.) In fact, if only short term contracts are of interest, then a better agreement among the curves in the first two subplots of Figure 10 can Inverse Problems and Imaging
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Figure 10. Implied (Black-Scholes) volatility corresponding to the local volatility surfaces, obtained with the six method variants (Tikhonov, EnKF and “no a0 ” applied to real and completed data) and compared to the market one. be obtained by simply dropping the data columns related to long term contracts! This is what some practitioners do in such a case, and we have verified that it works to significantly improve agreement among the curves, although not necessarily our state of knowledge about the actual financial implications. In order to assess the misfit between our reconstructed results and the practical implied volatility in our methods, we make use of three different figures of merit. Note that our inverse problem solution process is not directly trying to minimize such quantities, and thus they should be only taken as an additional quality control L ba quantity. Let Ii,j (Ii,j ) denote the implied volatility corresponding to the reconstructed local volatility (from the average of bid and ask option prices) with strike Ki and maturity τj . In the SPX example, we restrict the strikes to be between 1890 and 2500. Further, Vi,j is the volume of the option price with strike Ki and maturity τj , and Nvol is the number of contracts that have a nonzero volume. We Inverse Problems and Imaging
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define RM SE
=
sX L − I ba )2 /N (Ii,j vol , i,j
(root mean square error)
i,j
RW M SE
sX L − I ba )2 × V /N = (Ii,j i,j vol , i,j
(root weighted mean square error)
i,j
RR
=
sX sX L − I ba )2 / ba )2 , (Ii,j (Ii,j i,j i,j
(relative residual).
i,j
The resulting values are presented in Table 6. Table 6. Measures of data misfit of the 6 models. Scarce RMSE 0.0195 RWMSE 0.0175 RR 0.1407
Tikhonov-type EnKF Comp. Scarce (no a0 ) Comp. (no a0 ) Scarce Comp. 0.0321 0.0290 0.0325 0.0255 0.0324 0.0241 0.0252 0.0242 0.0241 0.0242 0.1987 0.2292 0.2186 0.1766 0.2186
Discussion of the equity data results. From the above experiment we conclude that using real or completed data sets we get quite different results. Within the completed data set, if we discard the a0 penalty, the two wings of the local volatility surface are not stable, for both methods of Sections 2 and 3. This is apparent in all the figures as well as Table 5. From the results involving reconstruction of the implied volatility as shown in Table 6, the Tikhonov-type method using the original (scarce) data has the best residuals in all three measures, with the EnKF method a close second. 5.3. Results for real data from commodities and data completion. Commodities have been traded extensively in different markets throughout the world for centuries. In many of those markets, a number of liquid options on such assets are also traded. Here again, data from such markets are freely available, and modelling such data is very relevant for financial analysts and risk management applications. In the present set of examples, we consider the adaption of Dupire’s model to the context of option prices on commodity futures introduced in [3]. For the present purpose, it consists of essentially the model presented in Sections 1 and 2. We chose data from WTI4 futures and their options, as well as Henry Hub5 contracts. The end-of-the-day WTI option and future prices were traded on 06Sep-2013, with the maturity dates, 18-Oct-2013, 16-Nov-2013, 17-Dec-2013, 16Jan-2014, 15-Feb-2014 and 20-Mar-2014. The end-of-the-day Henry Hub option and future prices were traded on 06-Sep-2013, with the maturity dates, 29-Oct2013, 27-Nov-2013, 27-Dec-2013, 29-Jan-2014, 26-Feb-2014 and 27-Mar-2014. The option prices were divided by their underlying future prices, so S0 = 1. In what follows, by residual we mean the merit function (7) with Γ = α0−1 I, α0 > 0. The penalty function (10) was used with α1 = α2 = α3 . A gradient descent method was applied in the minimization of the resulting Tikhonov-type 4 West
Texas Intermediate (WTI) is a grade of crude oil used as a benchmark in oil pricing. Hub (HH) natural gas futures are standardized contracts traded on the New York Mercantile Exchange (NYMEX). 5 Henry
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regularization functional. The parameter α0 was chosen by a discrepancy-based method inspired by the classical Morozov principle. See [3] for further details.
Figure 11. Reconstructed local volatility for different maturity dates for Henry Hub call option prices, comparing between completed data (green line with pentagram) and scarce data (blue line) results.
Figure 12. Reconstructed local volatility for different maturity dates for WTI call option prices, comparing between completed data (green line with pentagram) and scarce data (blue line) results.
To complete the data (when this alternative was used in this subsection) we applied linear interpolation, taking into account the boundary and initial conditions in (4). The boundary conditions were applied at y = ±5. When using completed data, we evaluated the local volatility only in the maturity times given in the original data, and interpolated it linearly in time, at each step in the minimization. Also, whenever |y| > 0.5 was encountered, we set a(τ, y) = a(τ, 0.5 ∗ sign(y)). When using the given scarce data, at each iteration, for each maturity time τ in the data set, we interpolated a(τ, ·) linearly in the intervals [−5, ymin ), and (ymax , 5], assuming that a(τ, −5) = max{0.08, a(τ, ymin )} and a(τ, 5) = max{0.08, a(τ, ymax )}, where ymin and ymax are the minimum and maximum log-moneyness strikes in the data set corresponding to τ . Figures 11 and 12 display reconstructed local volatility surfaces for the different maturities, comparing between using the given data and the completed data. Figures 13 and 14 present a comparison between the implied volatilities of both methods and the market ones, in order to assess how accurate the reconstructions are. One of the main advantages of the local volatility model is the capability of fitting the market implied smile, which has an important relationship with market risk. The implied volatilities were evaluated using the Matlab function blsimpv, and we used the interest rate as the dividend yield. Inverse Problems and Imaging
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Figure 13. Henry Hub prices: completed data (green line with pentagram), scarce data (blue continuous line), and market (red squares) implied volatilities.
21
Figure 14. WTI prices: completed data (green line with pentagram), sparse data (blue continuous line), and market (red squares) implied volatilities.
In all of these experiments, we have used the mesh widths ∆τ = 1/365 and ∆y = 0.05, the annualized risk-free interest rate was taken as 0.25%, and b = 0 in (3), since futures have no drift. Table 7 displays the parameters obtained in the tests of local volatility calibration with Henry Hub and WTI call prices, with scarce and completed data. In this table, by residual, we mean the `2 -distance between the evaluated quantity and the data, normalized by the `2 -norm of the data. Table 7. Parameters obtained in the local volatility calibration with Henry Hub and WTI call prices using sparse data and completed data.
α0 α1 = α2 = α3 Price Residual Implied Vol. Residual
WTI Comp. Data Sparse Data 1.0e4 1.0e3 4.5 1.0 2.16e-2 3.21e-3 1.26e-1 2.66e-2
Henry Hub Comp. Data Sparse Data 1.0e3 1.0e3 1.3 1.0 3.47e-2 2.14e-2 9.61e-2 5.98e-2
Discussion of the real commodity data results. Observing the market implied volatilities and the implied volatilities obtained with both methods in Figures 13 and 14, the results with scarce data present a much better smile adherence than when using completed data. So, completing the data can be seen as an unnecessary introduction of noise or inconsistency. It can be better noticed when observing the implied volatilities at deep in-the-money (y < −0.1) and deep out-of-the-money Inverse Problems and Imaging
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(y > 0.1) strikes. In these regions, for almost all cases, the results with scarce data practically matched the implied volatility, whereas with data completion, the resulting implied volatilities presented higher values. For financial market practitioners, higher implied volatilities can be translated to higher risk. So, using data completion could lead investors to be more conservative than necessary. 6. Conclusions. The questions of how to treat observed data in order to produce agreeable solutions, and relatedly, how much to trust the quality of a given data set collected by another agent, are prevalent in many nontrivial applied inverse problems. Standard theory appears to have little to contribute to their satisfactory resolution, and more carefully assembled experience is required. In this article we have highlighted some of the issues involved through an important application in mathematical finance, and we have proposed methods that improve on techniques available in the open literature. The problem of constructing a local volatility surface has similarities with some applications in areas such as geophysics and medical imaging, in that it boils down to the calibration of a diffusive PDE problem, reconstructing a distributed parameter function, i.e., a surface, rather than a few unrelated parameter values. The difficulty of dealing with scarce data, which highlights the need for a careful practical selection of a prior, is common, as is the uncertainty in data location, although the latter appears here as dependence only on a scalar additional variable. The finance problem considered here is distinguished from its more physical comrades in two important aspects. One is the availability of real data: we have experimented here with three different real applications, while most papers appearing in the applied mathematics literature never get to deal with real data at all. The other aspect is the difficulty of assessing the resulting recovered volatilities: there is no physical solution here, data sets are changing daily, and experience rules. In the present setting we had to determine the coefficients of the regularization operators, for instance αi in (8), (10) and (15), by trial and error. We remark that we are not proposing to get rid of synthetic data altogether: see, in particular, Section 5.1. But we are encouraging care. In fact, in between the two options of real sparse data and synthetic data everywhere there is the option of using synthetic data at the locations where real data are available. This option is employed routinely in exploration geophysics research, for instance. In general practice, though, synthetic results still tend to “look better” compared to those for real data. Of particular interest is the regularization term involving a0 and α1 . This term is a penalty on the sought local volatility surface for straying away from a given constant, a0 , which in turn is estimated based on the type of asset under consideration. The question whether or not this should be done is a deep one and touches upon the very foundations of the model under consideration. It also has a practical unfolding, since the possibility that the volatility, as a function of the asset price, grows at a sufficiently fast rate may be connected (at least in some similar models) to the presence of market bubbles [21]. The EnKF method considered in Section 3 is an adaptation of one of several methods proposed in the literature [20, 27], improved by adding smoothing penalty terms. We have also experimented with a similar adaptation of the method in [8]. In both cases the results are not consistently better than those obtained by the Tikhonov-type method of Section 2, which in hindsight is not surprising. However, Inverse Problems and Imaging
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since such methods are currently very much in vogue (especially in our target application area) we feel that our results in this sense are important: one must have data of sufficient quality for such methods to dominate.
Appendix A. EnKF details. The regularized weighted least-squares problem (13) has the solution a ˜h
=
(D−1 + H T Γ−1 H + LTτ Dτ−1 Lτ + LTy Dy−1 Ly )−1 · (D−1 a ˆ0 + H T Γ−1 d + LTτ Dτ−1 Lτ a ˆ0 + LTy Dy−1 Ly a ˆ0 ).
(19)
However, D is unknown. To address this we apply EnKF and replace D by the covariance matrix computed from generated samples. We also replace Lτ a ˆ0 and Ly a ˆ0 in (13) by rτ and ry , sampled from N (Lτ a ˆ0 , Dτ ) and N (Ly a ˆ0 , Dy ), respectively. In this way, rτ and ry can be viewed as two observations like d. Next we explain the calculations in one iteration of the prediction and analysis steps that appear in the EnKF algorithm stated at the end of Section 3. (0,j) (0,j) Having generated {ah }Jj=1 as the initial ensemble, we calculate P uh (ah ) and define ! (0,j) ah (1,j) a ˆh = , j = 1, 2, . . . , J. (0,j) P uh (ah ) This is the first prediction step (i.e., we set n = 0 in the stated algorithm). The sample mean and covariance matrix are then given by D1 =
J 1 X (1,j) (1,j) T (1) (1) T a ˆ (ˆ ah ) − a ¯h (¯ ah ) , J j=1 h
(1)
a ¯h =
J 1 X (1,j) a ˆ . J j=1 h
We now calculate the Kalman gain three times to obtain (D1−1 + H T Γ−1 H + LTτ Dτ−1 Lτ + LTy Dy−1 Ly )−1 . Denoting U = (D1−1 + H T Γ−1 H + LTτ Dτ−1 Lτ )−1 , we have (D1−1 + H T Γ−1 H + LTτ Dτ−1 Lτ + LTy Dy−1 Ly )−1 = (U −1 + LTy Dy−1 Ly )−1 = U − U LTy (Ly U LTy + Dy )−1 Ly U = (I − W1 Ly )U, where W1 = U LTy (Ly U LTy + Dy )−1 . We continue with the same procedure to calculate U . Let V −1 = D1−1 + H T Γ−1 H and W2 = V LTτ (Lτ V LTτ + Dτ )−1 . Then we have U
=
(D1−1 + H T Γ−1 H + LTτ Dτ−1 Lτ )−1 = (V −1 + LTτ Dτ−1 Lτ )−1
= V − V LTτ (Lτ V LTτ + Dτ )−1 Lτ V = (I − W2 Lτ )V. Hence, defining also W3 = D1 H T (HD1 H T + Γ)−1 we have V
=
(D1−1 + H T Γ−1 H)−1 = D1 − D1 H T (HD1T + Γ)−1 HD1
=
(I − W3 H)D1 .
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Now (19) reads (1,j)
a ˜h
(1,j) (1,j) (1,j) = U D−1 a ˆh + H T Γ−1 d + LTτ Dτ−1 Lτ a ˆh + LTy Dy−1 Ly a ˆh (1,j) (1,j) = U U −1 a ˆh + H T Γ−1 (d − Hˆ ah ) (1,j) (1,j) + LTτ Dτ−1 (rτ − Lτ a ˆh ) + LTy Dy−1 (ry − Ly a ˆh ) (1,j) (1,j) (1,j) = a ˆh + U H T Γ−1 (d − Hˆ ah ) + LTτ Dτ−1 (rτ − Lτ a ˆh ) (1,j) + LTy Dy−1 (ry − Ly a ˆh ) .
Reversing this procedure and adjusting notation, we obtain the analysis step 2(b) of the algorithm in Section 3 for n = 0. In each iteration of the EnKF algorithm, the prediction step is used to map the samples into the data space. The analysis step then calculates the “distance” between the mapped ensemble and the noisy data. Following the analysis step, the approximated local variance and estimated option prices are calculated, and this is used to compare with the original data. We compute the residual and compare to the one from the last iteration: Step 2 of the algorithm is repeated until the residual is less than a certain threshold. REFERENCES [1] Y. Achdou and O. Pironneau, Computational Methods for Option Pricing, SIAM, 2005. [2] Y. Achdou and O. Pironneau, Numerical procedure for calibration of volatility with American options, Applied Mathematical Finance, 12 (2007), 201–241. [3] V. Albani, U. Ascher and J. Zubelli, Local volatility models in commodity markets and online calibration, J. Computational finance, 2017. Accepted, to appear. [4] V. Albani and J. P. Zubelli, Online local volatility calibration by convex regularization, Appl. Anal. Discrete Math., 8 (2014), 243–268. [5] U. Ascher, H. Huang and K. van den Doel, Artificial time integration, BIT , 47 (2007), 3–25. [6] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Pol. Econ., 81 (1973), 637–654. [7] P. Boyle and D. Thangaraj, Volatility estimation from observed option prices, Decisions in Economics and Finance, 23 (2000), 31–52. [8] D. Calvetti, O. Ernst and E. Somersalo, Dynamic updating of numerical model discrepancy using sequential sampling, Inverse Problems, 30 (2014), 114019, 19pp. [9] A. De Cezaro, O. Scherzer and J. Zubelli, Convex regularization of local volatility models from option prices: convergence analysis and rates, Nonlinear Analysis, 75 (2012), 2398–2415. [10] A. De Cezaro and J. P. Zubelli, The tangential cone condition for the iterative calibration of local volatility surfaces, IMA Journal of Applied Mathematics, 80 (2015), 212–232. [11] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18–20. [12] H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates, Inverse Problems, 21 (2005), 1027–1045. [13] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, 1996. [14] J. Gatheral, The Volatility Surface: A Practitioner’s Guide, Wiley Finance. John Wiley & Sons, 2006. [15] J. Granek and E. Haber, Data mining for real mining: A robust algorithm for prospectivity mapping with uncertainties, Proc. SIAM Conference on Data Mining, (2015), 9pp. [16] E. Haber, U. Ascher and D. Oldenburg, Inversion of 3D electromagnetic data in frequency and time domain using an inexact all-at-once approach, Geophysics, 69 (2004), 1216–1228. Inverse Problems and Imaging
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[17] B. Hofmann and R. Kr¨ amer, On maximum entropy regularization for a specific inverse problem of option pricing, J. Inverse Ill-Posed Problems, 13 (2005), 41–63. [18] B. Hofmann, B. Kaltenbacher, C. P¨ oschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987–1010. [19] H. Huang and U. Ascher, Fast denoising of surface meshes with intrinsic texture, Inverse Problems, 24 (2008), 034003, 18pp. [20] M. Iglesias, K. Law and A. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 045001, 20pp. [21] R. Jarrow, Y. Kchia and P. Protter, How to detect an asset bubble, SIAM J. Financial Mathematics, 2 (2011), 839–865. [22] C. Johns and J. Mandel, A two-stage ensemble Kalman filter for smooth data assimilation, Environmental and Ecological Statistics, 15 (2008), 101–110. [23] N. Kahale, Smile interpolation and calibration of the local volatility model, Risk Magazine, 1 (2005), 637–654. [24] R. Korn and E. Korn, Option Price and Portfolio Optimization: Modern Methods of Mathematical Finance, volume 31 of Graduate Studies in Mathematics, AMS, 2001. [25] R. Kumar, C. da Silva, O. Aklain, A. Aravkin, H. Mansour, B. Recht and F. Herrmann, Efficient matrix completion for seismic data reconstruction, Geophysics, 80 (2015), 97–114. [26] G. Nakamura and R. Potthast, Inverse Problems. An Introduction to the Theory and Methods of Inverse Problems and Data Assimilation, IOP Publishing, 2015. [27] S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge, 2015. [28] F. Roosta-Khorasani, K. van den Doel and U. Ascher, Data completion and stochastic algorithms for PDE inversion problems with many measurements, ETNA, 42 (2014), 177–196. [29] C. Vogel, Computational Methods for Inverse Problem, SIAM, Philadelphia, 2002.
Received September 2016; revised May 2017. E-mail E-mail E-mail E-mail
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